On the weak coupling limit of the periodic quantum Lorentz gas

The Lorentz gas is an important model in kinetic theory where one seeks to under- stand the behavior of a single particle interacting with a fixed environment. In certain scaling regimes, the complicated microscopic behavior of the particle can be well-approximated by an effective equation or stochastic process. A description of the effective behavior usually does not rely on all details of the microscopic dynamics. When the interactions are quantum mechanical, the model is referred to as the quantum Lorentz gas. In this thesis, we study the quantum Lorentz gas in a periodic, weakly-coupled environment. We show that for certain observables, the limiting effective behavior in this regime is given by a transport equation, while for certain other observables, we provide evidence that the existence of the limit hinges on regularity properties of a certain phase space distribution that we introduce.